The present invention relates to the determination of the resonant frequency and/or Q of a device such as a gyro.
Angular rate sensors are used as components of navigational and inertial guidance systems for aircraft, spacecraft, ships, missiles, etc. Although mechanical gyroscopes have been used in the past for angular rate sensing, ring laser gyros and vibrating quartz gyros have displaced mechanical gyros because ring laser gyros and vibrating quartz gyros have characteristics that are superior to those of mechanical gyros.
A particularly economical vibrating quartz gyro employs pairs of parallel tines. Such a quartz gyro is described, for example, in Fersht et al., U.S. Pat. No. 5,056,366 and in Staudte, U.S. Pat. No. Re 32,931. One pair of tines (the drive tines) is driven by an oscillator so that the tines move toward and away from each other. Rotational motion of the tines about a central longitudinal axis causes the vibration of the drive tines to couple, by coriolis force, to the other pair of tines (the pick-off tines). The coriolis force causes the pick-off tines to vibrate in such a way that, when one pick-off tine moves in one direction, another pick-off tine moves in the opposite direction. The force, which drives the pick-off tines, is proportional to the cross product of the angular rate of rotation and the linear velocity of the drive tines.
The angular rate of rotation of the quartz gyro about the sensing axis appears as a double-sideband suppressed-carrier (DSSC) modulation of the input angular rate, where the carrier frequency is the frequency of oscillation of the drive tines. Therefore, an angular-rate signal can be recovered from the pick-off signal by a synchronous demodulator.
Analog circuits have been used for exciting the quartz gyro and for synchronous demodulation of the pick-off signal. Analog circuits, however, are subject to voltage offsets and drift of component values due to temperature variations and aging. These problems are particularly troublesome due to the peculiarities of the quartz gyro that are not apparent from the simplified or xe2x80x9cfirst orderxe2x80x9d operating characteristics of the gyro. One problem is related to the resonant frequencies of the drive tines and the pick-off tines. On the one hand, it is undesirable for the pick-off tines to have the same resonant frequency as the drive tines because of the difficulty of removing the dynamics of the pick-off tines from the pick-off signal. That is, if the pick-off tines have the same resonant frequency as the drive tines, the angular-rate signal would be a very non-linear function of the angular rate. On the other hand, the resonant frequency of the pick-off tines must be tuned relatively closely to the resonant frequency of the drive tines, or else the dynamic range of the angular-rate signal is limited by noise. Therefore, some resonant frequency offset between the drive tines and the pick-off tines is desirable.
A typical quartz gyro for inertial navigation applications, for example, has a drive resonant frequency of about 10 kilohertz, a Q of about 18,000, and a difference of about 100 Hz between the drive resonant frequency and the pick-off resonant frequency. The pick-off tines, for example, have the higher resonant frequency. This difference in resonant frequencies causes the amplitude of the angular-rate signal to be dependent on the frequency as well as on the amplitude of vibration of the drive tines.
As can be seen from the above, the resonant frequency of the gyro is set according to the desired specifications for the gyro. For example, it is known to set the resonant frequency of a gyro by laser trimming the tines until the desired resonant frequency is attained. It is helpful to the setting of the resonant frequency of a gyro for the resonant frequency and Q of the gyro to be quickly and accurately determined.
The present invention is directed to an arrangement for permitting the resonant frequency and Q of a gyro to be determined.
In accordance to one aspect of the present invention, a method of determining a resonant frequency of a device comprises the following: energizing the device with a short burst of energy to produce a decaying sinusoidal output; determining coefficients of a linear difference equation characterizing the decaying sinusoidal output by use of a least-mean-square fit; and, determining the resonant frequency from the coefficients.
In accordance with another aspect of the present invention, a method comprises the following: delaying an input by one sample period to produce a first delayed quantity; delaying the input by two sample periods to produce a second delayed quantity; squaring the second delayed quantity; averaging the squared second delayed quantity to produce a first output P0; multiplying the first and second delayed quantities to produce a first product; averaging the first product to produce a second output P1; squaring the first delayed quantity; subtracting a square of a first sample of the input from the squared first delayed quantity to form a first difference; dividing the first difference by a total number of samples of the input to produce a first result; adding the first output P0 to the first result to produce a third output P2; multiplying the input and the first delayed quantity to produce a second product; subtracting a product of the first sample and a second sample of the input from the second product to form a second difference; dividing the second difference by the total number of samples of the input to produce a second result; adding the second output P1 to the second result to produce a fourth output R0; multiplying the input and the second delayed quantity to produce a third product; averaging the third product to produce a fifth output R1; and, determining a resonant frequency f of a device providing the input based upon the first, second, third, fourth, and fifth outputs.
In accordance with still another aspect of the present invention, a method comprises the following: delaying an input by one sample period to produce a first delayed quantity; delaying the input by two sample periods to produce a second delayed quantity; squaring the second delayed quantity; averaging the squared second delayed quantity to produce P0; combining an expected value S0 of the square of a noise sample with P0 to produce a first output Pxe2x80x20; multiplying the first and second delayed quantities to produce a first product; averaging the first product to produce P1; combining an expected value S1 of a product of two adjacent noise samples with P1 to produce a second output Pxe2x80x21; squaring the first delayed quantity; subtracting a square of a first sample of the input from the squared first delayed quantity to form a first difference; dividing the first difference by a total number of samples of the input to produce a first result; adding the first output Pxe2x80x20 to the first result to produce a third output Pxe2x80x22; multiplying the input and the first delayed quantity to produce a second product; subtracting a product of the first sample and a second sample of the input from the second product to form a second difference; dividing the second difference by the total number of samples of the input to produce a second result; adding the second output Pxe2x80x21 to the second result to produce a fourth output Rxe2x80x20; multiplying the input and the second delayed quantity to produce a third product; averaging the third product to produce R1; combining an expected value S2 of a product of two noise samples separated by one sampling period with R1 to produce a fifth output Rxe2x80x21; and, determining a resonant frequency f of a device providing the input based upon the first, second, third, fourth, and fifth outputs.
In accordance with yet another aspect of the present invention, a method of determining a resonant frequency of a gyro comprises the following: energizing the gyro with a short burst of energy to produce a decaying sinusoidal output from the gyro; sampling the decaying sinusoidal output with a sampling period T to produce samples xn; determining coefficients a and b of a linear difference equation characterizing the decaying sinusoidal output by use of a least-mean-square fit, wherein the linear difference equation has the following form:
xn+2+axn+1+bxn=0 
wherein the least-mean-square fit used to determine the coefficients a and b is in accordance with the following equation:             min                                    a            ^                    m                ,                              b            ^                    m                      ⁢                  J        m            ⁢              xe2x80x83            ⁢      where      ⁢              xe2x80x83            ⁢              J        m              =            ∑              n        =        0            m        ⁢                  (                              x                          n              +              2                                +                                                    a                ^                            m                        ⁢                          x                              n                +                1                                              +                                                    b                ^                            m                        ⁢                          x              n                                      )            2      
where xc3xa2m and {circumflex over (b)}m are LMS estimates of coefficients a and b based upon m+2 input samples, and wherein the coefficients a and b of the linear difference equation are estimated by taking a partial derivative of Jm with respect to xc3xa2m and a partial derivative of Jm with respect to {circumflex over (b)}m, by equating both partial derivatives to zero, and by solving for the coefficients xc3xa2m and {circumflex over (b)}m; and, determining the resonant frequency from the estimates of the coefficients a and b.
In accordance with a further aspect of the present invention, a method of determining a resonant frequency f of a gyro comprises the following: energizing the gyro with a short burst of energy to produce a decaying sinusoidal output signal from the gyro; sampling the decaying sinusoidal output signal with a sampling period T to produce samples xn; estimating coefficients a and b of the decaying sinusoidal output signal from the samples xn; and, determining an estimate {circumflex over (ƒ)}m of the resonant frequency f from the estimates of the coefficients a and b according to the following equation:             f      ^        m    =            1              2        ⁢        π        ⁢                  xe2x80x83                ⁢        T              ⁢                            cos                      -            1                          ⁡                  (                                    -                                                a                  ^                                m                                                    2              ⁢                                                                    b                    ^                                    m                                                              )                    .      